Amplitude and Phase Fluctuations for Gravitational Waves Propagating through Inhomogeneous Mass Distribution in the Universe

Transcription

1 Amplitude and Phase Fluctuations fo Gavitational Waves Popagating though Inhomogeneous Mass Distibution in the Univese Ryuichi Takahashi Nagoya Univ. PD RT ApJ RT, uyama & Michikoshi A&A 438 L5 005

2 Abstact GW popagates though inhomogeneous mass distibution in the univese density petubation of CDM eve souce The amplitude and phase of GW ae changed by scatteing o lensing by density petubation. We deive these fluctuations based on a linea petubation theoy in the wave optics.

3 Intoduction Cosmological tandad ien The distance to the binay D is diectly detemined by its chip signal f f A A : amplitude 3 f D Lensing Eos A D A D A 1/ D Holz & Hughes 005; Kocsis et al. 006 The amplitude is changed by the lensing magnification 1/ o demagnification D D D D 1 seveal % in weak lensing Batelmann & chneide 001

4 edshift eo Eos on the infeed edshift to a LIA coalescence event, fo MBH binay M UN weak lensing eo intinsic LIA eo edshift Weak lensing eos lagely eceed othe souces of uncetainties. Kocsis et al. 006

5 Pevious wok Holz & Hughes 005; Kocsis et al. 006 The geometical optics was assumed. Only the amplitude is changed by lensing magnification, but the phase is not changed. This wok We use the wave optics. Not only the amplitude but also the phase is changed. We discuss lensing effects on both of them. If the scale of the density petubation is smalle than the Fesnel scale 100pc F 1/ f mhz, one should use the wave optics. Macquat 004; RT, uyama & Michikoshi 005

6 Lensed Wavefom petubed FRW metic ds a B d g d d 1 U d 1 U : confomal time a : scale facto U : gavitational potential wave equation fo scala field g B g 0 B We use the scala wave, instead of GW, since both wave equations ae same., 4 U Petes 1974 : Fouie component

7 Gavitational Lens Geomety : comoving distance z d z z H z 0 : two-dimensional vecto pependicula to, : incident wave : scatteed wave 1 0

8 4 U 1 0 L, U e d i Wave equation Lensed wavefom 0 1 incident wave scatteed wave spheical wave Including the effect of U on the fist ode. Bon appoimation

9 lensed wavefom at eve L 0 1 L Let s define K and as 0 i 1 K e geometical optics limit K 1 1 K Re & Im 0 0 : convegence t d : time delay K : amplitude fluctuation K>0 : magnification K<0 : demagnification : phase fluctuation : phase shift due to the lensing Ishimau 1978

10 Amplitude Fluctuation, 3, 0 0 a H U Vaiance in the amplitude fluctuation,,, k F k P k d a d H K K K Poisson s eq. Filte function Eisenstein & Hu 1999 ; Peacock & Dodds 1996 linea non-linea density petubation chaacteized by CDM powe spectum Fluctuation of the gavitational potential

11 Filte function Fesnel scale F K sin k F F k k : wave numbe of density petubation F 1/ f / 10pc mhz 10Gpc 1/ k 4 F GW does not epeience density petubation of scale less than Fesnel scale Macquat 004; RT, uyama & Michikoshi 005

12 Results ms amplitude fluctuation geometical optics souce edshift 10 K deceases fo f 10 Hz

13 Phase Fluctuation in chip signal Phase shift d t fo ince time delay does not change the wavefom, we use t d as the phase fluctuation Δ : ms phase diffeence fom ω1 to ω t t 1, 1 1 d d In the chip signal, the wavelength deceases GW feels the density petubation of the smalle scale

14 Hz Results ms phase diffeences Δ in chip signal fom f1 to f Hz f f z souce edshift z 3 z Phase fluctuation is typically seveal in unit of adian 10 3 ad. Phase can not be measued with an accuacy less than these values.

15 imple eample Density fluctuation is confined on a thin lens plane. U, k k souce eve Phase fluctuation t d C / 10 3 ad C : constant

16 Conclusion We conside GW popagating though a inhomogeneous mass distibution in the Univese, and discuss the lensing effects on the amplitude and the phase of a wavefom ms amplitude fluctuation is 1-10% fo Hz and <5% fo Hz.. ms phase fluctuation is 10 3 ad fo LIA fequency band. The phase can not measued with an accuacy less than this value. 3. Measuements of these fluctuations will povide infomation about the density fluctuation of 100pc scale.

17 Fesnel scale F Pass diffeence between pass 1 and is wave length λ Pass 1 souce F. Pass

18 Measuement accuacy fo the phase Ψ In the matched filteing analysis, the phase can be measued to an accuacy N ad 3 N /10 1 Hence if N > 10, the phase fluctuation becomes impotant and the phase can be detemined with an accuacy of Δ. N : signal-to-noise atio 1 N N